Structural, elastic and electronic properties of tetragonal HfO2 at applied hydrostatic pressure up to 50 GPa have been investigated using the plane-wave ultrasoft pseudopotential technique based on the first-principles densityfunctional theory (DFT). The calculated ground-state properties are in good agreement with previous theoretical and experimental data. Six independent elastic constants of tetragonal HfO2 have been calculated at zero pressure and high pressure. From the obtained elastic constants, the bulk, shear and Young's modulus, Poisson's coefficients, acoustic velocity and Debye temperature have been calculated at the applied pressure. Band structure shows that tetragonal HfO2 is an indirect band gap. The variation of the gap versus pressure is well fitted to a quadratic function.